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Find Energy from canonical function

  1. Aug 23, 2014 #1
    1. The problem statement, all variables and given/known data

    Let Z=2T/s + 1/3 + s/(30T) be the partition function of a quantum rotor at s/T->0. Show that

    [itex]U=nk(T-s/6-s^2/(180T)[/itex]

    2. Relevant equations

    1/(1+x) = 1 -x

    3. The attempt at a solution

    [itex]U=-kT^2(\partial _{T} ln(Z) )[/itex]

    Hence
    [itex]U=-kT^2 [2/s - s/(30T^2)] / [ 2T/s + 1/3 + s/(30T) ][/itex]

    I tried dividing upper and lower equation by 2T/s and use 1/(1+x) = 1 -x but cannot find the result.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 23, 2014 #2

    TSny

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    Are you sure the negative sign on the right side is correct?


    Try factoring out 2T/s in the denominator.

    Note that you want to get a result for U that is good to second order in the small quantity s/T.

    So, you might need to include the next higher order in the approximation 1/(1+x) = 1 - x
     
  4. Aug 28, 2014 #3

    nrqed

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    I think that it should be [itex]U=nkT^2(\partial _{T} ln(Z) )[/itex]

    I got their answer. You are doing it right but, as already pointed out, maybe you forgot to keep the expansion to second order, which you have to do here,
    [itex] 1/(1-x) = 1 + x + x^2 + \ldots [/itex]
     
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